KC_n的分解与K[X]上不可约多项式的对应关系

Corresponding Between Decomposition of KCn and Irreducible Polynomials on K[x]

本文主要证明当K是有理数域Q的扩域,Cn是n阶循环群时,正则模KCn分解为不可约KCn—模的直和与多项式xn-1分解为K[x]上不可约多项式的乘积之间的一一对应关系。对每个直和因子V,计算出HomKCn（V,V）的具体结构,以及利用上述模分解与多项式分解的对应关系证明当标量域K作有限正规扩张时,对应不可约直和因子必裂成若干维数相等且互不同构的直和因子。

In this paper We prove that when K is an extension of the rational field Q and Cn is a cylic group, there is a one to one corresponding between the decomposition of regular module KC into direct sums of irreducible left KCn-modules and the factorization of x^n-1 on K[x].For every irreducible component V,We calculate the structure of HomKC （V,V）.By using of this kind of corresponding, We also prove that when L/K is Galois extension, every irreducible component of KCn,if not absolutely irreducible, must split into some new components under the field extension such that they have the same dimension as irreducible LCn--modules.